July 03, 2018 Tuesday
Bedtime Story
Understanding Riemann's idea of Space
Tonight we shall continue with the 1854
paper of Bernhard Riemann that defined the modern idea of manifold space.
“The reason of this is doubtless that the
general notion of multiply extended magnitudes (in which space-magnitudes) are
included, remained entirely unworked.
I have in the first place, therefore, set
myself the task of constructing the notion of a multiply extended magnitude out
of general notions of magnitude.
It will follow from this that a multiply
extended magnitude is capable of different measure-relations, and consequently
that space is only a particular case of a triply extended magnitude.
But hence flows as a necessary consequence
that the propositions of geometry cannot be derived from general notions of
magnitude, but that the properties which distinguish space from other
conceivable triply extended magnitudes are only to be deduced from experience.”
Do not be distraught if you find these
lines incomprehensible for Riemann had a mathematical imagination and intuition
that has little to compare with those of average apes.
William Kingdon Clifford, an English
mathematician who translated the paper of Riemann that I have been quoting
gives perhaps a more reconcilable explanation of the assertions of Riemann.
What Riemann is asserting in this paper is
that there are different kinds of lines and surfaces and so there must be
different kind of space of three dimensions.
It is only through our experience that we
can find out to which of these kinds of space in which we live belongs.
In particular, the axioms of plane geometry
are true within the limits of experiment on the surface of a sheet of paper,
and yet we know that the sheet of paper is really covered with a number of
ridges and furrows, upon which (the total curvature not being zero) these
axioms are not true.
Similarly, he says although the axioms of
solid geometry are true within the limits of experiment for finite portions of
our space, yet we have no reason to conclude that they are true for very small portions;
and if any help can be got thereby for the explanation of physical phenomena, we
may have reason to conclude that they are not true for very small portions of
space.
In fact, if told without proper context,
Riemann’s ideas seem extremely speculative even today and no wonder they were
largely ignored save by the English mathematician Clifford, who not only
translated Riemann’s work but used them as preface to his own work on non-Euclidian
Clifford space.
Harold Edwards who is an American mathematician
and the author of the book Riemann’s Zeta function accepts that Riemann’s style
is very difficult for most average mortals to grasp and yet any simplification,
paraphrasing or reworking of Riemann’s carries with it the risk of missing an
important idea, of obscuring a point of view which was a source of Riemann’s
insight, or of introducing new technicalities or side issues which are not of
real concern.
I am not yet done with this remarkable
paper yet.
I have so far only quoted the introductory
part of the paper (along with some commentary on it) that runs to about eleven
pages and there is something else that I would like to add on it in the nights
to come.
Stay tuned to the voice of an average story storytelling
chimpanzee or login at http://panarrans.blogspot.com
Good night Mon Ami and my fellow cousin ape.
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Another great educator and a teacher that I am aware of is
Professor Subhashish Chattopadhyay in Bangalore, India.
While I narrate stories, Professor Subhashish an electronic
engineer and a former professor at BARC, does and teaches real mathematics and
physics.
He started the participation of Indian students at the
International Physics Olympiad.
Do visit him here:
All his books can be downloaded for free through this link:
For edutainment and English education of your children, I
recommend this large collection of Halloween Songs for Kids:
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